
// Make newform 5070.2.a.s in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5070_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_5070_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5070_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_5070_a();"
function MakeCharacter_5070_a()
    N := 5070;
    order := 1;
    char_gens := [1691, 4057, 1861];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_5070_a_Hecke(Kf)
    return MakeCharacter_5070_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[1], [-1], [1], [0], [0], [0], [-6], [0], [-4], [-10], [0], [6], [-2], [-4], [0], [-6], [0], [6], [-4], [-16], [2], [0], [-4], [6], [-14], [6], [12], [-4], [14], [10], [-12], [12], [-6], [4], [14], [16], [-14], [4], [-24], [-14], [-20], [-10], [24], [-14], [22], [-16], [-12], [-16], [-20], [-2], [18], [-16], [14], [-4], [-6], [-28], [14], [24], [2], [6], [-12], [-26], [4], [-16], [26], [-2], [-8], [18], [12], [-34], [26], [-24], [20], [10], [-24], [8], [-18], [14], [-18], [-2], [12], [22], [24], [2], [-24], [-4], [14], [-6], [14], [24], [-28], [0], [-8], [20], [40], [-36], [38], [10], [-20], [22], [28], [30], [-28], [-6], [-4], [-22], [12], [-6], [24], [42], [-36], [22], [-30], [-40], [48], [2], [-4], [-36], [-6], [-20], [30], [42], [-30], [-12], [-32], [-18], [-2], [24], [12], [-50], [40], [48], [24], [2], [22], [-34], [-26], [36], [18], [26], [-32], [22], [-44], [12], [54], [0], [38], [-38], [-20], [-48], [14], [34], [52], [-36], [28], [40], [-16], [46], [-22], [-10], [36], [10], [32], [12], [2], [-16], [-8], [-22], [-2], [42], [-56], [-10], [48], [58], [-16], [-54], [0], [-2], [-52], [22], [32], [-52], [-46], [2], [8], [38], [-14], [-4], [46], [48], [-22], [52], [36], [6], [-52], [34], [38], [-46], [-54], [44], [46], [40], [22], [-62], [-32], [-38], [-16], [-60], [22], [68], [50], [-10], [-44], [-4], [-24], [70], [44], [18], [-64], [-50], [14], [-40], [-26], [64], [28], [-50], [-6], [40], [28], [-32], [-62], [52], [-24], [-70], [52], [0], [-58], [-42], [-60], [-24], [-60], [36], [-20], [-10], [66], [0], [-32], [-16], [-40], [-4], [-2], [6], [-48], [58], [-46], [24], [-2], [-20], [-6], [34], [44], [-4], [-34], [-6], [-70], [-12], [30], [30], [-44], [18], [6], [-68], [26], [-72], [42], [-32], [-76], [-10], [-34], [-60], [60], [-40], [-36], [46], [-84], [-48], [-54], [6], [-8], [18], [-50], [-60], [-6], [40], [74], [54], [0], [2], [-36], [12], [-86], [-18], [88], [-36], [20], [-22], [12], [-10], [8], [10], [-28], [-18], [-78], [76], [88], [-70], [-56], [56], [10], [-30], [-60], [18], [30], [-72], [26], [50], [-16], [-20], [52], [-6], [-18], [10], [-56], [-28], [48], [76], [-66], [-6], [14], [68], [54], [74], [6], [-32], [-66], [36], [6], [-36], [-40], [-14], [80], [34], [-34], [44], [54], [-46], [8], [-32], [66], [48], [6], [42], [84], [-24], [12], [-30], [14], [56], [-38], [4], [20], [-24], [-34], [-32], [-54], [32], [-48], [26], [18], [-46], [-90], [42], [-24], [66], [-80], [48], [8], [-46], [-52], [-52], [46], [-42], [-16], [-44], [-8], [66], [8], [26], [84], [-90], [-66], [-6], [80], [18], [-18], [0], [-34], [78], [-44], [-8], [58], [-22], [-28], [-28], [-46], [6], [-104], [-52], [-6], [-12], [30], [-58], [-48], [-36], [18], [-18], [-76], [-2], [80], [56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5070_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5070_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_5070_2_a_s(:prec:=1)
    chi := MakeCharacter_5070_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_5070_2_a_s();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_5070_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5070_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![0, 1]>,<17,R![6, 1]>,<31,R![0, 1]>],Snew);
    return Vf;
end function;